I am trying to understand an example of semidirect products that I read in Dummit and Foote. This is the example of the classification of groups of order 12 in page 182 and 183. In this case we consider that $G$ is a group of order 12, $V \in Syl_2(G)$ and $T\in Syl_3(G)$. My first question comes when we are assuming $V\trianglelefteq G$and $V \cong Z_2 \times Z_2$. I understand why we have 1 trivial homomorphism from $T \rightarrow Aut(V)$ and 2 non-trivial.
Q1) Why, when we consider these two non-trivial homomorphisms, do we get the group $A_4$? I know that the books makes a reference to a previous example, but I still don't understand where we get this group $A_4$ from. Can anyone explain this?
Also, I have a question when we assume $T \trianglelefteq G$. Particularly, I have a question when we are assuming $V \cong Z_2 \times Z_2$. I can see why we have three nontrivial homomorphisms.
Q2) What I am failing to see is why the resulting three semidirect products are all isomorphic to $S_3 \times Z_2$, can anyone explain this?
Q3) In general, I am having trouble, once I identify the possible homomorphisms into the automorphism of some subgroup of a group, identifying what groups the resulting semidirect product could be. Can anyone suggest some way to see this, or give any hint on how to do it? Thanks so much for your help!